Mahler's method and Carlitz logarithm

TL;DR

This paper offers a new proof of Papanikolas's theorem on the algebraic independence of Carlitz logarithms by employing Mahler's method, providing an alternative to the t-motives approach.

Contribution

It introduces a Mahler's method-based proof for Papanikolas's theorem, extending Denis's approach to a broader class of algebraic functions.

Findings

New proof of algebraic independence using Mahler's method

Extension of Denis's approach to general algebraic functions

Alternative perspective on Carlitz logarithm independence

Abstract

In 2007, Papanikolas established that if Carlitz logarithms of algebraic functions are linearly independent over the rational function field, then they are algebraically independent. The purpose of the present paper is to provide a new proof of this theorem using Mahler s method instead of the theory of t-motives. We revisit and extend the approach developed by Denis, which enabled him in 2006 to prove this result in the particular case of the logarithm of elements in Fq(theta) via a Mahler system.